prompt listlengths 1 1 | reward_model dict | data_source stringclasses 4
values | ability stringclasses 2
values | extra_info dict |
|---|---|---|---|---|
[
{
"content": "Find the auxiliary equation for the ordinary differential equation with constant coefficients: \\((x^2D^2 + xD + 1)y = \\sin(2\\log x)\\sin(\\log x)\\).",
"role": "user"
}
] | {
"ground_truth": "m^2 + 1 = 0",
"style": "rule-lighteval/MATH_v2"
} | deepmath_filtered | MATH | {
"index": "0",
"problem": "Find the auxiliary equation for the ordinary differential equation with constant coefficients: \\((x^2D^2 + xD + 1)y = \\sin(2\\log x)\\sin(\\log x)\\).",
"solution": "Okay, so I need to find the auxiliary equation for the given ordinary differential equation (ODE) with constant coeffi... |
[
{
"content": "Find the limit: \\[ \\lim_{x \\to \\infty} (x!)^{1/x} \\]",
"role": "user"
}
] | {
"ground_truth": "\\infty",
"style": "rule-lighteval/MATH_v2"
} | deepmath_filtered | MATH | {
"index": "1",
"problem": "Find the limit: \\[ \\lim_{x \\to \\infty} (x!)^{1/x} \\]",
"solution": "Okay, so I need to find the limit as x approaches infinity of (x!)^(1/x). Hmm, factorials can be tricky because they grow really fast, but maybe I can use some approximations or known formulas for factorials when ... |
[
{
"content": "Find the length of the polar curve given by \\( r = \\sqrt{1 + \\cos(2\\theta)} \\) for \\( 0 \\leq \\theta \\leq \\frac{\\pi\\sqrt{2}}{4} \\).",
"role": "user"
}
] | {
"ground_truth": "\\dfrac{\\pi}{2}",
"style": "rule-lighteval/MATH_v2"
} | deepmath_filtered | MATH | {
"index": "2",
"problem": "Find the length of the polar curve given by \\( r = \\sqrt{1 + \\cos(2\\theta)} \\) for \\( 0 \\leq \\theta \\leq \\frac{\\pi\\sqrt{2}}{4} \\).",
"solution": "Okay, so I need to find the length of the polar curve given by \\( r = \\sqrt{1 + \\cos(2\\theta)} \\) from \\( \\theta = 0 \\)... |
[
{
"content": "Consider a regular pentagon circumscribed in a circle. Connect each vertex of this pentagon to every other non-adjacent vertex with a straight line segment to form a pentagram, which contains a smaller pentagon. What is the ratio of the area of the original (large) pentagon to the smaller one in t... | {
"ground_truth": "\\phi^4",
"style": "rule-lighteval/MATH_v2"
} | deepmath_filtered | MATH | {
"index": "3",
"problem": "Consider a regular pentagon circumscribed in a circle. Connect each vertex of this pentagon to every other non-adjacent vertex with a straight line segment to form a pentagram, which contains a smaller pentagon. What is the ratio of the area of the original (large) pentagon to the smalle... |
[
{
"content": "Determine the values of \\( k \\) for which the equation \\( x^3 - 3x^2 + 6x + k = 0 \\) has three real roots.",
"role": "user"
}
] | {
"ground_truth": "\\emptyset",
"style": "rule-lighteval/MATH_v2"
} | deepmath_filtered | MATH | {
"index": "4",
"problem": "Determine the values of \\( k \\) for which the equation \\( x^3 - 3x^2 + 6x + k = 0 \\) has three real roots.",
"solution": "Okay, so I need to figure out for which values of k the equation \\( x^3 - 3x^2 + 6x + k = 0 \\) has three real roots. Hmm, let's start by recalling that a cubi... |
[
{
"content": "Verify that the distance from the function \\( f = x \\) to the set \\( Y = \\{ f \\in C[0,1] : \\int_0^1 f = 0 \\} \\) is 0.5, given that \\( Y \\) is a closed subset of \\( X = \\{ f \\in C[0,1] : f(0) = 0 \\} \\).",
"role": "user"
}
] | {
"ground_truth": "0.5",
"style": "rule-lighteval/MATH_v2"
} | deepmath_filtered | MATH | {
"index": "5",
"problem": "Verify that the distance from the function \\( f = x \\) to the set \\( Y = \\{ f \\in C[0,1] : \\int_0^1 f = 0 \\} \\) is 0.5, given that \\( Y \\) is a closed subset of \\( X = \\{ f \\in C[0,1] : f(0) = 0 \\} \\).",
"solution": "Okay, so I need to verify that the distance from the f... |
[
{
"content": "Let $a, b, c, d$ be a permutation of the numbers $1, 9, 8, 4$. Define $n = (10a + b)^{10c + d}$. Calculate the probability that $1984!$ is divisible by $n$. Use Fermat's Little Theorem to assist in your calculations.",
"role": "user"
}
] | {
"ground_truth": "\\dfrac{5}{6}",
"style": "rule-lighteval/MATH_v2"
} | deepmath_filtered | MATH | {
"index": "6",
"problem": "Let $a, b, c, d$ be a permutation of the numbers $1, 9, 8, 4$. Define $n = (10a + b)^{10c + d}$. Calculate the probability that $1984!$ is divisible by $n$. Use Fermat's Little Theorem to assist in your calculations.",
"solution": "Alright, so I need to solve this problem where we have... |
[
{
"content": "Find parametric equations for a unit circle with a speed of \\( e^t \\), starting from \\( x=1 \\), \\( y=0 \\). Determine when the circle is completed.",
"role": "user"
}
] | {
"ground_truth": "\\ln(2\\pi + 1)",
"style": "rule-lighteval/MATH_v2"
} | deepmath_filtered | MATH | {
"index": "7",
"problem": "Find parametric equations for a unit circle with a speed of \\( e^t \\), starting from \\( x=1 \\), \\( y=0 \\). Determine when the circle is completed.",
"solution": "Okay, so I need to find parametric equations for a unit circle that starts at (1,0) and has a speed of e^t. Then I hav... |
[
{
"content": "Using the generating function \\( \\sum_{n=0}^{\\infty}P_n(x)r^n=(1-2rx+r^2)^{-\\frac{1}{2}} \\), find the value of \\( P_n(1) \\).",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | deepmath_filtered | MATH | {
"index": "8",
"problem": "Using the generating function \\( \\sum_{n=0}^{\\infty}P_n(x)r^n=(1-2rx+r^2)^{-\\frac{1}{2}} \\), find the value of \\( P_n(1) \\).",
"solution": "Okay, so I need to find the value of \\( P_n(1) \\) using the generating function given by \\( \\sum_{n=0}^{\\infty}P_n(x)r^n = (1 - 2rx + ... |
[
{
"content": "Evaluate the expression \\( \\log \\left| 1 + \\alpha + \\alpha^2 + \\alpha^3 - \\frac{1}{\\alpha} \\right| \\), where \\( \\alpha \\) is a fifth root of unity.",
"role": "user"
}
] | {
"ground_truth": "\\log 2",
"style": "rule-lighteval/MATH_v2"
} | deepmath_filtered | MATH | {
"index": "9",
"problem": "Evaluate the expression \\( \\log \\left| 1 + \\alpha + \\alpha^2 + \\alpha^3 - \\frac{1}{\\alpha} \\right| \\), where \\( \\alpha \\) is a fifth root of unity.",
"solution": "Okay, so I need to evaluate the expression \\( \\log \\left| 1 + \\alpha + \\alpha^2 + \\alpha^3 - \\frac{1}{\... |
[
{
"content": "Find the kernel of the Vandermonde matrix \\( A \\) given by\n\\[\nA = \\begin{pmatrix} x_1^0 & x_1^1 & \\ldots & x_1^n \\\\ x_2^0 & x_2^1 & \\ldots & x_2^n \\\\ \\vdots & \\vdots & \\ldots & \\vdots \\\\ x_m^0 & x_m^1 & \\ldots & x_m^n \\end{pmatrix}\n\\]\nwhere \\( n < m - 1 \\) and the \\( x_i ... | {
"ground_truth": "\\{\\mathbf{0}\\}",
"style": "rule-lighteval/MATH_v2"
} | deepmath_filtered | MATH | {
"index": "10",
"problem": "Find the kernel of the Vandermonde matrix \\( A \\) given by\n\\[\nA = \\begin{pmatrix} x_1^0 & x_1^1 & \\ldots & x_1^n \\\\ x_2^0 & x_2^1 & \\ldots & x_2^n \\\\ \\vdots & \\vdots & \\ldots & \\vdots \\\\ x_m^0 & x_m^1 & \\ldots & x_m^n \\end{pmatrix}\n\\]\nwhere \\( n < m - 1 \\) and t... |
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